<div style="text-align:justify;"> In order to speed up the global optimization-based mesh smoothing, an enhanced steepest descent method is presented in the paper. Numerical experiment results show tha...<div style="text-align:justify;"> In order to speed up the global optimization-based mesh smoothing, an enhanced steepest descent method is presented in the paper. Numerical experiment results show that the method performs better than the steepest descent method in the global smoothing. We also presented a physically-based interpretation to explain why the method works better than the steepest descent method. </div>展开更多
The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent m...The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities is introduced. Strong convergence of this method is established under suitable assumptions imposed on the algorithm parameters.展开更多
Vertical records are critically important when determining the rupture model of an earthquake, especially a thrust earthquake. Due to the relatively low fitness level of near-field vertical displacements, the precisio...Vertical records are critically important when determining the rupture model of an earthquake, especially a thrust earthquake. Due to the relatively low fitness level of near-field vertical displacements, the precision of previous rupture models is relatively low, and the seismic hazard evaluated thereafter should be further updated. In this study, we applied three-component displacement records from GPS stations in and around the source region of the 2013 MW6.6 Lushan earthquake to re-investigate the rupture model.To improve the resolution of the rupture model, records from both continuous and campaign GPS stations were gathered, and secular deformations of the GPS movements were removed from the records of the campaign stations to ensure their reliability. The rupture model was derived by the steepest descent method（SDM）, which is based on a layered velocity structure. The peak slip value was about 0.75 m, with a seismic moment release of 9.89 × 1018 N·m, which was equivalent to an MW6.6 event. The inferred fault geometry coincided well with the aftershock distribution of the Lushan earthquake. Unlike previous rupture models, a secondary slip asperity existed at a shallow depth and even touched the ground surface. Based on the distribution of the co-seismic ruptures of the Lushan and Wenchuan earthquakes, post-seismic relaxation of the Wenchuan earthquake, and tectonic loading process, we proposed that the seismic hazard is quite high and still needs special attention in the seismic gap between the two earthquakes.展开更多
In this paper,we address interesting soliton resolution,asymptotic stability of N-soliton solutions and the Painleve asymptotics for the Kundu-Eckhaus(KE)equation with nonzero boundary conditions iq_(t)+q_(xx)-2(l|q|^...In this paper,we address interesting soliton resolution,asymptotic stability of N-soliton solutions and the Painleve asymptotics for the Kundu-Eckhaus(KE)equation with nonzero boundary conditions iq_(t)+q_(xx)-2(l|q|^(2)-1)q+4β^(2)(lql^(4)-1)q+4iβ(lql^(2))_(x)q=0,q(x,0)=q_(0)(x)-±1,x→±∞.The key to proving these results is to establish the formulation of a Riemann-Hilbert(RH)problem associated with the above Cauchy problem and find its connection with the RH problem of the defocusing NLS equation.With the■-steepest descent method and the results of the defocusing NLS equation,we find complete leading order approximation formulas for the defocusing KE equation on the whole(x,t)half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region,Zakharov-Shabat asymptotics in a solitonless region and the Painlevéasymptotics in two transition regions.展开更多
In this paper, we consider the generalized variational inequality GVI（F, g, C）, where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iter...In this paper, we consider the generalized variational inequality GVI（F, g, C）, where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI（F,g, C）. Strong convergence results are established and applications to constrained generalized pseudo-inverse are included.展开更多
By extending the classical analysis techniques due to Samokish, Faddeev and Faddee- va, and Longsine and McCormick among others, we prove the convergence of the precon- ditioned steepest descent with implicit deflati...By extending the classical analysis techniques due to Samokish, Faddeev and Faddee- va, and Longsine and McCormick among others, we prove the convergence of the precon- ditioned steepest descent with implicit deflation （PSD-id） method for solving Hermitian- definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotie estimate of the rate of convergence of the PSD-id method. We show that with a proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal which leads to superlinear convergence Numerical examples are presented to verify the theoretical results on the convergence behavior of the PSD- id method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper shed light on an improved understanding of the convergence behavior of these block methods.展开更多
Spectral conjugate gradient method is an algorithm obtained by combination of spectral gradient method and conjugate gradient method,which is characterized with global convergence and simplicity of spectral gradient m...Spectral conjugate gradient method is an algorithm obtained by combination of spectral gradient method and conjugate gradient method,which is characterized with global convergence and simplicity of spectral gradient method,and small storage of conjugate gradient method.Besides,the spectral conjugate gradient method was proved that the search direction at each iteration is a descent direction of objective function even without relying on any line search method.Spectral conjugate gradient method is applied to full waveform inversion for numerical tests on Marmousi model.The authors give a comparison on numerical results obtained by steepest descent method,conjugate gradient method and spectral conjugate gradient method,which shows that the spectral conjugate gradient method is superior to the other two methods.展开更多
There are many phenomena that generate polygonal tessellations on surfaces of 3D objects. One interesting example is the jackfruit, a multiple fruit found in the tropics. A recent study found the best-fit spherical Vo...There are many phenomena that generate polygonal tessellations on surfaces of 3D objects. One interesting example is the jackfruit, a multiple fruit found in the tropics. A recent study found the best-fit spherical Voronoi diagram from a photo of jackfruit skin, but the optimization was relative to the radius of the sphere and the height of the spikes. In this study, we propose a method for adjusting the position of the center of the sphere in addition to these parameters. Experiments were conducted using both ideal and real data. However, convergence with real data has not been confirmed due to relaxation of the convergence condition.展开更多
The authors study the Cauchy problem for the focusing nonlinear KunduEckhaus(KE for short)equation and construct the long time asymptotic expansion of its solution in fixed space-time cone with C(x_(1),x_(2),v_(1),v_(...The authors study the Cauchy problem for the focusing nonlinear KunduEckhaus(KE for short)equation and construct the long time asymptotic expansion of its solution in fixed space-time cone with C(x_(1),x_(2),v_(1),v_(2))={(x,t)∈R^(2):x=x_(0)+vt,x_(0)∈[x_(1),x_(2)],v∈[v_(1),v_(2)]}.By using the inverse scattering transform,Riemann-Hilbert approach and δ steepest descent method,they obtain the lone time asymptotic behavior of the solution,at the same time,they obtain the solitons in the cone compare with the all N-soliton the residual error up to order O(t-^(3/4)).展开更多
We introduce the fractional-order global optimal backpropagation machine,which is trained by an improved fractionalorder steepest descent method(FSDM).This is a fractional-order backpropagation neural network(FBPNN),a...We introduce the fractional-order global optimal backpropagation machine,which is trained by an improved fractionalorder steepest descent method(FSDM).This is a fractional-order backpropagation neural network(FBPNN),a state-of-the-art fractional-order branch of the family of backpropagation neural networks(BPNNs),different from the majority of the previous classic first-order BPNNs which are trained by the traditional first-order steepest descent method.The reverse incremental search of the proposed FBPNN is in the negative directions of the approximate fractional-order partial derivatives of the square error.First,the theoretical concept of an FBPNN trained by an improved FSDM is described mathematically.Then,the mathematical proof of fractional-order global optimal convergence,an assumption of the structure,and fractional-order multi-scale global optimization of the FBPNN are analyzed in detail.Finally,we perform three(types of)experiments to compare the performances of an FBPNN and a classic first-order BPNN,i.e.,example function approximation,fractional-order multi-scale global optimization,and comparison of global search and error fitting abilities with real data.The higher optimal search ability of an FBPNN to determine the global optimal solution is the major advantage that makes the FBPNN superior to a classic first-order BPNN.展开更多
In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the ba...In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation.Furthermore,using the approach of Deift-Zhou nonlinear steepest descent,the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived.For the reverse space-time nonlocal Hirota equation,since the symmetries of its scattering matrix are different with the local Hirota equation,the v(λ_(i))(i=0,1)would like to be imaginary,which results in theδ_(λi)^(0)contains an increasing t(±Imv(λ_(i)))/2,and then the asymptotic behavior for nonlocal Hirota equation becomes differently.展开更多
文摘<div style="text-align:justify;"> In order to speed up the global optimization-based mesh smoothing, an enhanced steepest descent method is presented in the paper. Numerical experiment results show that the method performs better than the steepest descent method in the global smoothing. We also presented a physically-based interpretation to explain why the method works better than the steepest descent method. </div>
基金Project supported by the Key Science Foundation of Education Department of Sichuan Province of China (No.2003A081)Sichuan Province Leading Academic Discipline Project (No.SZD0406)
文摘The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space is studied. A new three-step relaxed hybrid steepest-descent method for this class of variational inequalities is introduced. Strong convergence of this method is established under suitable assumptions imposed on the algorithm parameters.
基金supported by the grant from the National Sichuan-Yunnan earthquake prediction experimental field of CEA (grant No. 2016CESE0204)
文摘Vertical records are critically important when determining the rupture model of an earthquake, especially a thrust earthquake. Due to the relatively low fitness level of near-field vertical displacements, the precision of previous rupture models is relatively low, and the seismic hazard evaluated thereafter should be further updated. In this study, we applied three-component displacement records from GPS stations in and around the source region of the 2013 MW6.6 Lushan earthquake to re-investigate the rupture model.To improve the resolution of the rupture model, records from both continuous and campaign GPS stations were gathered, and secular deformations of the GPS movements were removed from the records of the campaign stations to ensure their reliability. The rupture model was derived by the steepest descent method（SDM）, which is based on a layered velocity structure. The peak slip value was about 0.75 m, with a seismic moment release of 9.89 × 1018 N·m, which was equivalent to an MW6.6 event. The inferred fault geometry coincided well with the aftershock distribution of the Lushan earthquake. Unlike previous rupture models, a secondary slip asperity existed at a shallow depth and even touched the ground surface. Based on the distribution of the co-seismic ruptures of the Lushan and Wenchuan earthquakes, post-seismic relaxation of the Wenchuan earthquake, and tectonic loading process, we proposed that the seismic hazard is quite high and still needs special attention in the seismic gap between the two earthquakes.
基金supported by the National Science Foundation of China(Grant No.12271104,51879045)。
文摘In this paper,we address interesting soliton resolution,asymptotic stability of N-soliton solutions and the Painleve asymptotics for the Kundu-Eckhaus(KE)equation with nonzero boundary conditions iq_(t)+q_(xx)-2(l|q|^(2)-1)q+4β^(2)(lql^(4)-1)q+4iβ(lql^(2))_(x)q=0,q(x,0)=q_(0)(x)-±1,x→±∞.The key to proving these results is to establish the formulation of a Riemann-Hilbert(RH)problem associated with the above Cauchy problem and find its connection with the RH problem of the defocusing NLS equation.With the■-steepest descent method and the results of the defocusing NLS equation,we find complete leading order approximation formulas for the defocusing KE equation on the whole(x,t)half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region,Zakharov-Shabat asymptotics in a solitonless region and the Painlevéasymptotics in two transition regions.
文摘In this paper, we consider the generalized variational inequality GVI（F, g, C）, where F and g are mappings from a Hilbert space into itself and C is the fixed point set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI（F,g, C）. Strong convergence results are established and applications to constrained generalized pseudo-inverse are included.
文摘By extending the classical analysis techniques due to Samokish, Faddeev and Faddee- va, and Longsine and McCormick among others, we prove the convergence of the precon- ditioned steepest descent with implicit deflation （PSD-id） method for solving Hermitian- definite generalized eigenvalue problems. Furthermore, we derive a nonasymptotie estimate of the rate of convergence of the PSD-id method. We show that with a proper choice of the shift, the indefinite shift-and-invert preconditioner is a locally accelerated preconditioner, and is asymptotically optimal which leads to superlinear convergence Numerical examples are presented to verify the theoretical results on the convergence behavior of the PSD- id method for solving ill-conditioned Hermitian-definite generalized eigenvalue problems arising from electronic structure calculations. While rigorous and full-scale convergence proofs of preconditioned block steepest descent methods in practical use still largely eludes us, we believe the theoretical results presented in this paper shed light on an improved understanding of the convergence behavior of these block methods.
文摘Spectral conjugate gradient method is an algorithm obtained by combination of spectral gradient method and conjugate gradient method,which is characterized with global convergence and simplicity of spectral gradient method,and small storage of conjugate gradient method.Besides,the spectral conjugate gradient method was proved that the search direction at each iteration is a descent direction of objective function even without relying on any line search method.Spectral conjugate gradient method is applied to full waveform inversion for numerical tests on Marmousi model.The authors give a comparison on numerical results obtained by steepest descent method,conjugate gradient method and spectral conjugate gradient method,which shows that the spectral conjugate gradient method is superior to the other two methods.
基金Partly Supported by the Grant-in-Aid for Basic Research of MEXT(No.24360039)
文摘There are many phenomena that generate polygonal tessellations on surfaces of 3D objects. One interesting example is the jackfruit, a multiple fruit found in the tropics. A recent study found the best-fit spherical Voronoi diagram from a photo of jackfruit skin, but the optimization was relative to the radius of the sphere and the height of the spikes. In this study, we propose a method for adjusting the position of the center of the sphere in addition to these parameters. Experiments were conducted using both ideal and real data. However, convergence with real data has not been confirmed due to relaxation of the convergence condition.
文摘The authors study the Cauchy problem for the focusing nonlinear KunduEckhaus(KE for short)equation and construct the long time asymptotic expansion of its solution in fixed space-time cone with C(x_(1),x_(2),v_(1),v_(2))={(x,t)∈R^(2):x=x_(0)+vt,x_(0)∈[x_(1),x_(2)],v∈[v_(1),v_(2)]}.By using the inverse scattering transform,Riemann-Hilbert approach and δ steepest descent method,they obtain the lone time asymptotic behavior of the solution,at the same time,they obtain the solitons in the cone compare with the all N-soliton the residual error up to order O(t-^(3/4)).
基金Project supported by the National Key Research and Development Program of China(No.2018YFC0830300)the National Natural Science Foundation of China(No.61571312)。
文摘We introduce the fractional-order global optimal backpropagation machine,which is trained by an improved fractionalorder steepest descent method(FSDM).This is a fractional-order backpropagation neural network(FBPNN),a state-of-the-art fractional-order branch of the family of backpropagation neural networks(BPNNs),different from the majority of the previous classic first-order BPNNs which are trained by the traditional first-order steepest descent method.The reverse incremental search of the proposed FBPNN is in the negative directions of the approximate fractional-order partial derivatives of the square error.First,the theoretical concept of an FBPNN trained by an improved FSDM is described mathematically.Then,the mathematical proof of fractional-order global optimal convergence,an assumption of the structure,and fractional-order multi-scale global optimization of the FBPNN are analyzed in detail.Finally,we perform three(types of)experiments to compare the performances of an FBPNN and a classic first-order BPNN,i.e.,example function approximation,fractional-order multi-scale global optimization,and comparison of global search and error fitting abilities with real data.The higher optimal search ability of an FBPNN to determine the global optimal solution is the major advantage that makes the FBPNN superior to a classic first-order BPNN.
基金supported by the National Natural Science Foundation of China(No.12175069 and No.12235007)Science and Technology Commission of Shanghai Municipality(No.21JC1402500 and No.22DZ2229014)Natural Science Foundation of Shanghai(No.23ZR1418100)。
文摘In this work,we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector.Start from the Lax pair,we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation.Furthermore,using the approach of Deift-Zhou nonlinear steepest descent,the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived.For the reverse space-time nonlocal Hirota equation,since the symmetries of its scattering matrix are different with the local Hirota equation,the v(λ_(i))(i=0,1)would like to be imaginary,which results in theδ_(λi)^(0)contains an increasing t(±Imv(λ_(i)))/2,and then the asymptotic behavior for nonlocal Hirota equation becomes differently.
